Question

In Fig. $ ABC$ is a quadrant of a circle of radius $ 14 cm$ and a semicircle is drawn with $ BC$ as diameter. Find the area of the shaded region.

Answer 1

Step 1. Find area of quarter circle

The radius of the circle is $14cm$

$\therefore AB=AC=14cm$

Area of quarter circle$=\frac{1}{4}\mathrm{\pi }·{\mathrm{r}}^2\left[\mathrm{r}\mathrm{the}\mathrm{radius}\mathrm{of}\mathrm{circle}\right]$

$$\begin{gathered} =\frac{1}{4}\times \frac{22}{7}\times {14}^2 \\ =154c{m}^2 \end{gathered}$$

Step 2. Find area of triangle

Area of triangle is $=\frac{1}{2}\times \mathrm{Base}\times \mathrm{Height}$

$$\begin{gathered} =\frac{1}{2}\times AB\times AC \\ =\frac{1}{2}\times 14\times 14 \\ =98c{m}^2 \end{gathered}$$

Step 3. Find hypotenuse

From Pythagoras theorem we know

$$\begin{gathered} {\left(\mathrm{Hypoteneus}\right)}^2={\left(\mathrm{Height}\right)}^2+{\left(\mathrm{Base}\right)}^2 \\ \Rightarrow {\mathrm{BC}}^2={\mathrm{AB}}^2+{\mathrm{AC}}^2 \\ \Rightarrow \mathrm{BC}=\sqrt{{\mathrm{AB}}^2+{\mathrm{AC}}^2} \\ \Rightarrow \mathrm{BC}=\sqrt{{14}^2+{14}^2} \\ \Rightarrow \mathrm{BC}=14\sqrt{2} \end{gathered}$$

Step 4. Find area of half circle

The area of the half circle whose diameter is the hypotenuse is$=\frac{1}{2}\times \mathrm{\pi }\times {\left(\frac{\mathrm{Hypoteneus}}{2}\right)}^2$

$$\begin{gathered} =\frac{1}{2}\times \frac{22}{7}\times {\left(\frac{14\sqrt{2}}{2}\right)}^2 \\ =154c{m}^2 \end{gathered}$$

Step 5. Find area of shaded region

The area of the shaded region is $=\mathrm{Area}\mathrm{of}\mathrm{the}\mathrm{half}\mathrm{circle}-\left(\mathrm{Area}\mathrm{of}\mathrm{the}\mathrm{quarter}\mathrm{circle}-\mathrm{area}\mathrm{of}\mathrm{the}\mathrm{triangle}\right)$

$$\begin{gathered} =154-\left(154-98\right) \\ =98c{m}^2 \end{gathered}$$

Hence, the area of the shaded region is $98c{m}^2$.